The remaining deviation with respect to the exact evolution of QFI, which occurs both at equilibrium and at the center of the pump, can be attributed to the fact that trRIXS spectrum is not identical to the spin structure factor \(S(q,\omega ,t)\) . Such a discrepancy is known in equilibrium RIXS, which captures the poles of \(S(q,\omega ,t)\)  but is less accurate in yielding its spectral weight [1]}. Physically, the intermediate state in trRIXS has a finite lifetime and contains additional dynamics, such as multi-magnon or spin-charge excitations, besides instantaneous spin flip events in the UCL limit.
Due to the admixture with these excitations, Eq. REF  is only a good approximation but not an identity. The trRIXS spectrum underestimates the QFI at equilibrium, where the local moment is maximal and other processes are irrelevant (see the Fig. REF  in Appendix ), while overestimating the QFI at the center of the pump pulse where there are more charge carriers induced by the laser.
